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Randomness
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Randomness In Theory and Practice Alon Amit CMC3 Conference
Monterey, Dec 2014
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Randomness
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Randomness
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Randomness
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Randomness Mathematics
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1. Tournaments
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A
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Rule: every 3 players simultaneously lose to someone.
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Can this be done?
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YES.
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Actually…
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No.
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We need 100 Players.
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But How?
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No Algebra.
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No Geometry.
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No Rules.
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No Thought.
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Just choose each arrow at random.
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A
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Fail probability: 7/8
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All fail: (7/8)97
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Total Failed Tournaments:
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At least 2/3 of those random assignments succeed.
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The Probabilistic Method
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Sometimes it’s easier to randomly choose than to explicitly construct.
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Often, 99.9% of the objects have what you need…
…but you can’t find them! …
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2. High Girth and Chromatic Number
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Graphs
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Graphs A B C F E D Vertices
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Graphs A B C F E D Edges
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Graphs A B C F E D Vertices = Edges =
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Graphs A B C F E D Cycles
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Graphs A B C F E D Cycles
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Shortest Cycle = Girth = 3
Graphs A B C F E D Shortest Cycle = Girth = 3
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Graphs A B C F E D Coloring
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Graphs A B C F E D Chromatic Number = 3
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Can a graph have both high girth and large chromatic number?
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High girth: 2-colorable locally everywhere.
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How can we force many colors if every region is 2-colorable?
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That’s HARD.
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Unless…
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…you use the Probabilistic Method.
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Take many vertices Choose each edge with probability p Carefully choose p Make the graph have few short cycles and small independence number Remove vertices to eliminate short cycles Voila!
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3. The Existence of Designs
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7 points Sets of size 3 Every 2 points are in exactly 1 set
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7 points Sets of size 3 Every 2 points are in exactly 1 set
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n points Sets of size q Every r points are in exactly λ sets
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n points Sets of size q Every r points are in exactly λ sets
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? n q r λ
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? n q r λ Divisibility conditions Finitely many exceptions in n
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Asked: 1853
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Answered: Jan 15, 2014
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Solver: Peter Keevash
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Method: Randomized Algebraic Construction
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Rephrase the problem as hypergraph matching
Seek a matching by randomly picking edges and deleting their overlaps The beginning is easy, but the end is out of control Keevash: Cleverly pick stand-ins for the end game
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4. The Crossing Lemma
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Crossing Number: Minimum number of edge crossings in a plane drawing of a graph
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Crossing Lemma: if then
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Fact (easy): Start with any graph Pick a random subgraph H by choosing each vertex with probability p Find the expected number of vertices, edges and crossings of H Apply the easy fact. Pick the best p. Done!
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5. Algorithms
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Quicksort Codes Primality Testing Min-Cut Matrix Testing
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Machine Learning
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Machine Learning
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Machine Learning Learn from data: features and outcomes
Predict: Given features, what’s the outcome?
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Decision Trees
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Decision Trees are great…
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…Random Forests are Even Better.
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…Random Forests are Even Better.
Train 100 decision trees with random data and random features Merge into one predictor
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Perhaps the single most successful Machine Learning paradigm Ever.
Random Forests Perhaps the single most successful Machine Learning paradigm Ever.
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Summary
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Randomness Mathematics
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Artificial Intelligence
Randomness Mathematics Artificial Intelligence
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